Hybrid physics-based machine learning models open avenues for modeling systems that are not entirely understood or that require a large amount of data to be modeled. These models combine the high interpretability of high-fidelity models (HFMs) with the lower computational cost of machine learning models (MLs) while maintaining high accuracy [1, 2]. Based on that, this work proposes a generalizable framework for obtaining dynamic discrepancy reduced-order models (DD-ROMs) that balance the differences between HFMs and ROMs using Gaussian Processes (GPs), speeding up computational times for optimization and control while ensuring model's interpretability. The proposed framework is an evolution of discrepancy-based modeling approaches [3-6] and encompasses fundamental criteria [7] for addressing missing underlying physics. The framework offers a comprehensive step-by-step process, including: (i) where to place discrepancy terms guided by sensitivity and correlation analyses; (ii) how to obtain the discrepancy profiles for the parameters selected to be augmented; and (iii) how to train the discrepancy profiles based on the current system states and manipulated inputs using GPs. The proposed method is employed to correct dynamic mismatches between a reduced-order and a high-fidelity microkinetic model of the steam methane reforming (SMR) reaction. The results demonstrated that with the discrepancy function added to the equilibrium constant, the DD-ROM could mimic the dynamic trajectories of the microkinetic model with high accuracy, exhibiting an R2 of 99.42% and an RMSE of 0.08509. Additionally, the developed model provided computational gains, being 52 times faster per execution than integrating the HFM. This could be significant when considering that online dynamic real-time optimization and/or advanced control usually requires thousands of model executions to compute optimal profiles. The proposed framework will be extended to address missing underlying physics related to emerging technologies, such as carbon capture, utilization, and storage (CCUS) and electrochemical systems. In particular, a second case study focuses on developing a dynamic model for a CO2 electrochemical reduction (CO2RR) process using the DD-ROM approach. This model aims to be the first of its kind, bridging the gap between the dynamic aspects of electrochemical CO2 reduction processes and the underlying material and kinetic modeling related to them. Furthermore, the developed CO2RR model will be leveraged in future dynamic optimization studies driven by intermittent renewable energy profiles, targeting both techno-economic and environmental objectives.
REFERENCES
[1] Motagamwala, A.H., Dumesic, J.A., 2021. Microkinetic Modeling: A Tool for Rational Catalyst Design. Chem. Rev. 121, 1049–1076.
[2] Daoutidis, P., Lee, J.H., Rangarajan, S., Chiang, L., Gopaluni, B., Schweidtmann, A.M., Harjunkoski, I., Mercangöz, M., Mesbah, A., Boukouvala, F., Lima, F.V., Del Rio Chanona, A., Georgakis, C., 2024. Machine learning in process systems engineering: Challenges and opportunities. Computers & Chemical Engineering 181, 108523.
[3] Kennedy, M.C., O’Hagan, A., 2001. Bayesian Calibration of Computer Models. Journal of the Royal Statistical Society Series B: Statistical Methodology 63, 425–464.
[4] Mebane, D.S., Bhat, K.S., Kress, J.D., Fauth, D.J., Gray, M.L., Lee, A., Miller, D.C., 2013. Bayesian calibration of thermodynamic models for the uptake of CO2 in supported amine sorbents using ab initio priors. Phys. Chem. Chem. Phys. 15, 4355.
[5] Bhat, K.S., Mebane, D.S., Mahapatra, P., Storlie, C.B., 2017. Upscaling Uncertainty with Dynamic Discrepancy for a Multi-Scale Carbon Capture System. Journal of the American Statistical Association 112, 1453 -1467.
[6] Li, K., Mahapatra, P., Bhat, K.S., Miller, D.C., Mebane, D.S., 2017. Multi-scale modeling of an amine sorbent fluidized bed adsorber with dynamic discrepancy reduced modeling. React. Chem. Eng. 2, 550–560.
[7] Dinh S., Nascimento C.A., Mebane D.S., Lima F. V. A Framework for Implementation of Dynamic Discrepancy Reduced-Order Modeling. Submitted for publication. 2025.
